Skip to main content
SpringerLink
Log in

Branes and moduli spaces of Higgs bundles on smooth projective varieties

Research in the Mathematical Sciences Aims and scope Submit manuscript

Cite this article

Abstract

Given a smooth complex projective variety M and a smooth closed curve \(X\, \subset \, M\) such that the homomorphism of fundamental groups \(\pi _1(X)\, \longrightarrow \, \pi _1(M)\) is surjective, we study the restriction map of Higgs bundles, namely from the Higgs bundles on M to those on X. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on M and X. We also consider the setup where a finite group is acting on M via holomorphic automorphisms or anti-holomorphic involutions, and the curve X is preserved by this action. Branes are studied in this context.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Baraglia, D., Schaposnik, L.P.: Higgs bundles and \((A, B, A)\)-branes. Commun. Math. Phys. 331, 1271–1300 (2014)

    Article  MathSciNet  Google Scholar 

  2. Baraglia, D., Schaposnik, L.P.: Real structures on moduli spaces of Higgs bundles. Adv. Theor. Math. Phys. 20, 525–551 (2016)

    Article  MathSciNet  Google Scholar 

  3. Biswas, I.: Parabolic principal Higgs bundles. J. Ramanujan Math. Soc. 23, 311–325 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Biswas, I., Majumder, S., Wong, M.L.: Parabolic Higgs bundles and \(\Gamma \)-Higgs bundles. J. Aust. Math. Soc. 95, 315–328 (2013)

    Article  MathSciNet  Google Scholar 

  5. Chen, T.-H., Ngô, B.C.: On the Hitchin morphism for higher dimensional varieties. Duke Math. J. 169, 1971–2004 (2020)

    MathSciNet  MATH  Google Scholar 

  6. Corlette, K.: Flat G-bundles with canonical metrics. J. Differ. Geom. 28, 361–382 (1988)

    Article  MathSciNet  Google Scholar 

  7. Donaldson, S.K.: Twisted harmonic maps and the self-duality equations. Proc. Lond. Math. Soc. 55, 127–131 (1987)

    Article  MathSciNet  Google Scholar 

  8. Heller, S., Schaposnik, L.P.: Branes through finite group actions. J. Geom. Phys. 129, 279–293 (2018)

    Article  MathSciNet  Google Scholar 

  9. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59–126 (1987)

    Article  MathSciNet  Google Scholar 

  10. Hitchin, N.J.: Stable bundles and integrable systems. Duke Math. J. 54, 91–114 (1987)

    Article  MathSciNet  Google Scholar 

  11. Hitchin, N.J., Karlhede, A., Lindström, U., Rocek, M.: Hyperkähler metrics and supersymmetry. Commun. Math. Phys. 108, 535–589 (1987)

    Article  Google Scholar 

  12. Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1, 1–236 (2007)

    Article  MathSciNet  Google Scholar 

  13. Kobayashi, S.: Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15. Kanô Memorial Lectures, 5. Princeton University Press, Princeton (1987)

  14. Lamotke, K.: The topology of complex projective varieties after S. Lefschetz. Topology 20, 15–51 (1981)

    Article  MathSciNet  Google Scholar 

  15. Simpson, C.T.: Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization. J. Am. Math. Soc. 1, 867–918 (1988)

    Article  MathSciNet  Google Scholar 

  16. Simpson, C.T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 7(5), 5–95 (1992)

    Article  MathSciNet  Google Scholar 

  17. Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes Études Sci. Publ. Math. 79, 47–129 (1994)

  18. Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety. II, Inst. Hautes Études Sci. Publ. Math. 80, 5–79 (1994)

  19. Simpson, C.T.: The Hodge filtration on nonabelian cohomology, Algebraic geometry, Santa Cruz 1995, 217–281, Proc. Sympos. Pure Math. 62, Part 2, American Mathematical Society, Providence, RI (1997)

  20. Wentworth, R.A.: Higgs bundles and local systems on Riemann surfaces, Geometry and quantization of moduli spaces, 165–219. Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham (2016)

Download references

Acknowledgements

We thank the two referees for going through the paper very carefully. IB is supported by a J. C. Bose Fellowship. LPS is partially supported by NSF CAREER Award DMS-1749013. On behalf of all authors, the corresponding author states that there is no conflict of interest.

Author information

Authors and Affiliations

  1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005, India

    Indranil Biswas

  2. Institute of Differential Geometry, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Sebastian Heller

  3. Department of Mathematics Statistics, and Computer Science, University of Illinois at Chicago, 851 S Morgan St, Chicago, IL, 60607, USA

    Laura P. Schaposnik

Authors
  1. Indranil Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sebastian Heller
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Laura P. Schaposnik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Indranil Biswas.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Biswas, I., Heller, S. & Schaposnik, L.P. Branes and moduli spaces of Higgs bundles on smooth projective varieties. Res Math Sci 8, 52 (2021). https://doi.org/10.1007/s40687-021-00286-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40687-021-00286-z

Keywords

Mathematics Subject Classification

search

Navigation